First uncountable ordinal explained

In mathematics, the first uncountable ordinal, traditionally denoted by

\omega _{1} or sometimes by \Omega , is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. When considered as a set, the elements of \omega _{1} are the countable ordinals (including finite ordinals),^[1] of which there are uncountably many.

Like any ordinal number (in von Neumann's approach),

\omega _{1} is a well-ordered set, with set membership serving as the order relation. \omega _{1} is a limit ordinal, i.e. there is no ordinal \alpha such that \omega _{1} = \alpha +1 .

The cardinality of the set

\omega _{1} is the first uncountable cardinal number, \aleph _{1} (aleph-one). The ordinal \omega _{1} is thus the initial ordinal of \aleph _{1} . Under the continuum hypothesis, the cardinality of \omega _{1} is \beth _{1} , the same as that of R —the set of real numbers.^[2]

In most constructions,

\omega _{1} and \aleph _{1} are considered equal as sets. To generalize: if \alpha is an arbitrary ordinal, we define \omega _{\alpha} as the initial ordinal of the cardinal \aleph _{\alpha} .

The existence of

\omega _{1} can be proven without the axiom of choice. For more, see Hartogs number.

Topological properties

Any ordinal number can be turned into a topological space by using the order topology. When viewed as a topological space,

\omega _{1} is often written as [ 0 ,\omega _{1)} , to emphasize that it is the space consisting of all ordinals smaller than \omega _{1} .

If the axiom of countable choice holds, every increasing ω-sequence of elements of

[ 0 ,\omega _{1)} converges to a limit in [ 0 ,\omega _{1)} . The reason is that the union (i.e., supremum) of every countable set of countable ordinals is another countable ordinal.

The topological space

[ 0 ,\omega _{1)} is sequentially compact, but not compact. As a consequence, it is not metrizable. It is, however, countably compact and thus not Lindelöf (a countably compact space is compact if and only if it is Lindelöf). In terms of axioms of countability, [ 0 ,\omega _{1)} is first-countable, but neither separable nor second-countable.

The space

[ 0 ,\omega _{1]=\omega} _{1} + 1 is compact and not first-countable. \omega _{1} is used to define the long line and the Tychonoff plank—two important counterexamples in topology.

See also

• Epsilon numbers (mathematics)
• Large countable ordinal
• Ordinal arithmetic

References

1. Web site: Set Theory > Basic Set Theory (Stanford Encyclopedia of Philosophy). 2020-08-12. plato.stanford.edu.
2. Web site: first uncountable ordinal in nLab. 2020-08-12. ncatlab.org.

Bibliography

• Thomas Jech, Set Theory, 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, .
• Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology. Springer-Verlag, New York, 1978. Reprinted by Dover Publications, New York, 1995. (Dover edition).